Comput Mech (2011) 48:47–63DOI 10.1007/s00466-011-0580-y

ORIGINAL PAPER

Dispersion and transient analyses of Hermite reproducing kernelGalerkin meshfree method with sub-domain stabilized conformingintegration for thin beam and plate structures

Dongdong Wang · Zhenting Lin

Received: 14 November 2010 / Accepted: 31 January 2011 / Published online: 25 February 2011© Springer-Verlag 2011

Abstract A dispersion analysis is carried out to study thedynamic behavior of the Hermite reproducing kernel (HRK)Galerkin meshfree formulation for thin beam and plateproblems. The HRK approximation utilizes both the nodaldeflectional and rotational variables to construct the meshfreeapproximation of the deflection field within the reproducingkernel framework. The discrete Galerkin formulation is ful-filled with the method of sub-domain stabilized conformingintegration. In the dispersion analysis following the HRKGalerkin meshfree semi-discretization, both the deflection-al and rotational nodal variables are expressed by harmonicfunctions and then substituted into the semi-discretized equa-tion to yield the characteristic equation. Subsequently thenumerical frequency and phase speed can be obtained. Thetransient analysis with full-discretization is performed byusing the central difference time integration scheme. Theresults of dispersion analysis of thin beams and plates showthat compared to the conventional Gauss integration-basedmeshfree formulation, the proposed method has more favor-able dispersion performance. Thereafter the superior perfor-mance of the present method is also further demonstrated byseveral transient analysis examples.

Keywords Hermite reproducing kernel approximation ·Meshfree method · Thin beam and plate · Sub-domainstabilized conforming integration · Dispersion analysis ·Transient analysis

D. Wang (B) · Z. LinDepartment of Civil Engineering, Xiamen University,Xiamen 361005, Fujian, Chinae-mail: ddwang@xmu.edu.cn

1 Introduction

The meshfree methods [1–7] based on unstructured particlediscretrization of problem domain allows a straightforwardconstruction of arbitrary high order global conformingapproximation, i.e., the C1 approximation required for theclassical thin beam and plate problems. The reader maysee [8–13] for excellent summary about various types ofmeshfree methods and their recent developments. The salientadvantage of C1 meshfree approximation was first employedby Krysl and Belytschko for thin plate and shell analysis[14,15]. The meshfree free vibration analysis of thin beamand plate structures was also cried out by Liu et al. [16] andZhou et al. [17]. A meshless local Petrov Galerkin methodwas introduced by Long and Atluri [18] for thin plate bend-ing analysis. Rabczuk et al. [19] presented a meshfree thinshell formulation for nonlinear dynamic fracture simulation.In contrast to using a plate or shell formulation, the directthree dimensional meshfree analysis of plate or shell struc-tures was proposed as well by Li and Liu [20] and recentlyit was applied to predict the thin cylinder failure under thecombining thermal and mechanical loads by Qian et al. [21]and the dynamic fracture simulation of thin-walled structuresby Gato [22]. It is noticed that the aforementioned meth-ods solely utilize the translational degrees of freedom in thinbeam, plate or shell analysis. On the other hand the Hermitetypes of meshfree methods were also proposed for thin plateanalysis. Liu et al. [23–27] systematically presented a repro-ducing kernel element method in which the nodal rotationsare included in the defelctional approximation. A Hermite-type radial point interpolation method was also proposed byLiu et al. [28] for thin plate problems.

In this work the moving least square (MLS) [29] or repro-ducing kernel (RK) [4] approximation based Galerkin mesh-free method is particular referred. One key issue associated

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with the MLS/RK based Galerkin meshfree method is thathigher order Gauss quadrature rule is often required to per-form the domain integration owing to the rational feature ofthe MLS/RK meshfree shape functions. This issue becomeseven more pronounced for thin plate problems due to theintegration of the second order derivatives in the weak form.To improve the computational efficiency and simultaneouslyensure the spatial stability for Galerkin meshfree methods[30], a stabilized nodal integration method was developed byBeissel and Belytschko [31] via adding the residual squareof the equilibrium equation to the conventional potentialenergy functional. The gradient and dilatational stabilizationmethods with stress point integration were also discussed byDuan and Belytschko [32]. Starting from the linear exactnesscondition and the gradient smoothing technique [33], Chenet al. [34–36] introduced a stabilized conforming nodal inte-gration (SCNI) method which does not need any artificialparameters. Subsequently by employing the bending exact-ness condition this method was systematically developed foranalysis of the C0 beam, plate, and shell problems [37–43].More recently the SCNI method was further generalized toformulate the sub-domain stabilized conforming integration(SSCI) method for thin plate analysis in which a Hermitereproducing kernel (HRK) approximation was used [44,45].It was found that the HRK Galerkin meshfree formulationbased on the SSCI method works well for both static andfree vibration problems [45,46].

In this study the dispersion and transient analyses arecarried out to further assess the dynamic performance ofthe HRK Galerkin meshfree formulation based on the SSCIintegration methodology for thin beam and plate problems.Dispersion analysis is a very effective numerical tool for eval-uating the dynamic property of numerical schemes [47]. Inthe meshfree context, Christion and Voth [49] studied thedispersive properties of reproducing kernel semi-discretiza-tions. The dispersion analysis for the natural element methodwas presented by Bueche et al. [50]. Voth et al. [51] ana-lyzed the dispersive behavior for the advection problems withstabilized integration. The characteristics of semi- and full-discretizations of stabilized Galerkin meshfree method werepresented by You et al. [52]. Chen and Wu [53] investigatedthe Lagrangian and Semi-Lagrangian reproducing kernel dis-cretizations with nodal integration through dispersion anal-ysis. In this work for the HRK mesfree method with SSCI,the dispersion analysis is completed by expressing both thedefelctional and rotational nodal variables as series of har-monic functions. Subsequently the substitution of the nodalvector into the semi-discrete equation yields the characteris-tic equation that is used to extract the numerical frequency.The continuum frequency is also discussed in details andthe comparison of the numerical and continuum frequenciesgives the dispersive features of the present approach. Mean-while the transient analysis is also performed for the full-

discrete equation that is obtained by invoking the central dif-ference time integration method on the semi-discrete equa-tion. The comparison between the present SSCI method andother Gauss integration based methods manifests the superi-ority of the present approach.

The paper is organized as follows. Section 2 gives a briefsummary of the HRK meshfree approximation. In Sect. 3the SSCI thin plate formulation is discussed. The dispersionanalysis for the continuum as well as the semi-discrete prob-lems is presented in Sect. 4. Section 5 presents the fully dis-cretized equation for transient analysis. In Sect. 6 the resultsof dispersion and transient analyses are presented, which isfollowed the conclusions in Sect. 7.

2 HRK approximation

Consider a problem domain, i.e., the plate mid-plane �. Inthe HRK approximation [44–46], � is discretized by a set ofmeshfree particles {xI , xI ∈ �}NP

I=1, NP is the total numberof particles, the approximant of the field variable like theplate deflection w(x, t), say wh(x, t), takes the followingform:

wh(x, t) =∑

I∈H

[�w

I (x)wI (t) + �θxI (x)θx I (t)

+�θyI (x)θy I (t)

]=∑

I∈H� I (x)d I (t) (1)

where H = {I |supp(xI ) ⊃ x}, supp(xI ) denotes the sup-port of the kernel function �aI (xI − x), it is noted that�aI (xI − x) > 0 for x ∈ H and moreover one has� ∈ ∪N P

I=1[supp(xI )]. The support size aI of �aI (xI − x) isa distance measure in order to have �aI (xI − x) = 0 in casethat ‖xI − x‖ /aI ≥ 1. Here the conventional cubic B-splinekernel function is used [6]. dT

I (t) = {wI (t), θx I (t), θy I (t)}is the nodal deflectional and rotational coefficient vector,� I (x) = {�w

I (x),�θxI (x),�

θyI (x)}. �w

I (x),�θxI (x) and

�θyI (x) are the deflectional and rotational shape functions

associated with the meshfree node I, and they can beexpressed as:

�wI (x) = pT (xI − x)b(x)�aI (xI − x) (2)

�θxI (x) = pT

x (xI − x)b(x)�aI (xI − x) (3)

�θyI (x) = pT

y (xI − x)b(x)�aI (xI − x) (4)

where the vectors of p, px , and py are the monomial basisfunctions with orders n and (n −1), and b(x) is the unknownvector to be solved:

pT (x) = {1, x, y, x2, xy, y2, . . . , xn, . . . , yn} (5)

pTx (x) = {0, 1, 0, 2x, y, 0, . . . , nxn−1, . . . , 0} (6)

pTy (x) = {0, 0, 1, 0, x, 2y, . . . , 0, . . . , nyn−1} (7)

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Comput Mech (2011) 48:47–63 49

bT (x) = {b00(x), b10(x), b01(x), b20(x), b11(x),

b02(x), . . . , bn0(x), . . . , b0n(x)} (8)

Subsequently the following nth order reproducing condi-tion can be invoked to obtain the unknown vector b(x):∑

I∈H

[�w

I (x)p(xI − x) + �θxI (x)px (xI − x)

+ �θyI (x)py(xI − x)

]= p(0) (9)

Further substituting Eqs. (2)–(4) into Eq.(9) gives:

M(x)b(x) = p(0) (10)

where M(x) is the HRK moment matrix defined as:

M(x) = M0(x) + M1(x) (11)

with M0 and M1 being given by:

M0(x) =∑

I∈Hp(xI − x)pT (xI − x)�aI (xI − x) (12)

M1(x) =∑

I∈H

[px (xI − x)pT

x (xI − x)

+ py(xI − x)pTy (xI − x)

]�aI (xI − x) (13)

Thus from Eq. (10) we solve b(x) = M−1(x)p(0) and finallythe HRK shape functions of Eqs. (2)–(4) can be rewritten as:

�wI (x) = pT (0)M−1(x)p(xI − x)�aI (xI − x) (14)

�θxI (x) = pT (0)M−1(x)px (xI − x)�aI (xI − x) (15)

�θyI (x) = pT (0)M−1(x)py(xI − x)�aI (xI − x) (16)

It has been shown that the HRK meshfree shape functionshave better kernel stability compared to their standard RKcounterparts. For a quadratic basis function it has been showthat the normalized support size aI for HRK approximationis 1 while it is 2 for the conventional RK approximation[45,46]. Therefore the HRK approximation allows using asmaller support size for meshfree computation.

3 SSCI thin plate formulation

3.1 Field equations of thin plate

Consider a thin plate which occupies a domain B = � ×(−t/2, t/2), where � ∈ R

2 denotes the plate mid-planeenclosed by a boundary � and t is the plate thickness,respectively. According to the classical Kirchhoff hypoth-esis for thin plate, all the strain and stress measures can beexpressed by the transverse deflection of the middle surface:w(x), x = {x, y} ∈ �. The sign conventions of deflectionand rotations used in this paper are shown in Fig. 1. The

Fig. 1 Sign conventions for thin plate

equation of motion for a thin plate can be stated as:⎧⎨

⎩

ρtw − LT m − c = 0 in �

w = w; w,n = θn on �g

mnn = mnn; sn = sn on �h(17)

where ρ is the material density, the superposed doubledots over w denote the twice time differentiation, c isthe applied force per unit area of the plate mid-plane.LT = { ∂2/∂x2 ∂2/∂y2 2 × ∂2/∂x∂y }, sn = qn + mmn,s isthe modified shear force, m is the moment vector and basedon the isotropic linear constitutive relationship it is related tothe curvature vector κ = Lw as:

m =⎧⎨

⎩

mxx

myy

mxy

⎫⎬

⎭ = −Dκ; D = D

⎡

⎣1 υ 0υ 1 00 0 (1 − υ)/2

⎤

⎦ ;

D = Et3

12(1 − υ2)(18)

with E and υ being the Young’s modulus and Poisson ratio,respectively. The moment mnn and the shear force qn on thenatural boundary �h are given by:⎧⎪⎨

⎪⎩

qn = qx nx + qyny

mnn = mxx n2x + myyn2

y + 2mxynx ny

mns = (myy − mxx )nx ny + mxy

(n2

x − n2y

) (19)

The weak form that governs the motion of a thin plate canbe stated as follows:∫

�

δwρtw d� +∫

�

δκT DκT d� − δW ext = 0 (20)

with

δW ext =∫

�

δwc d� −∫

�h

δθnmnn d� +∫

�h

δwsn d�

+∑

C

δwCrC (21)

where the last term denotes the contribution of the force rC =mns(x

+C ) − mns(x

−C ) acting at a plate corner point xC , this

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50 Comput Mech (2011) 48:47–63

(a)

(b)

Fig. 2 Schematic illustration of curvature smoothing sub-domains fordomain integration: a SSCI-Q4; b SSCI-T8

term vanishes for a smooth boundary and for brevity it isignored in the subsequent discussion.

3.2 Sub-domain curvature smoothing and SSCI formulation

The essential ingredient of the SSCI formulation is the sub-domain curvature smoothing in which the nodal represen-tative domain is sub-divided into several sub-domains fordomain integration based on the smoothed curvature. It turnsout that this approach can meet the bending integration con-straint [44]. Figure 2 shows a schematic illustration of quadri-lateral and triangular smoothing or integration sub-domains,where �K denotes the nodal representative domain for theparticle xK and it is further sub-divided into N S non-over-lapping and conforming sub-domains, i.e.,

⋃N SKKm=1 �Km =

�K , N SK is the number of sub-domains associated withthe K th meshfree particle. The two type of sub-domains asshown in Fig. 2 are referred as SSCI-Q4 and SSCI-T8 [46] inthe following development, in case of one dimensional beamformulation, SSCI-Q4 and SSCI-T8 coincide and are termed“SSCI2”, which implies that for each nodal representative

Fig. 3 Schematic illustration of nodal relationship and wave propaga-tion direction

domain two sub-domains are used for curvature smoothingand domain integration.

In each sub-domain �Km , a smoothing curvature κ(xKm )

can be defined as:

κh(xKm ) = 1

AKm

∫

�Km

κh(x) d�= 1

AKm

∫

�Km

⎧⎪⎨

⎪⎩

wh,xx

wh,yy

2wh,xy

⎫⎪⎬

⎪⎭d�

= 1

AKm

∫

�Km

⎧⎪⎨

⎪⎩

wh,x nx

wh,yny

2wh,x ny

⎫⎪⎬

⎪⎭d�

= 1

AKm

∑

I∈Hm

⎡

⎢⎣∫

�Km

⎧⎨

⎩

� I,x nx

� I,yny

2� I,x ny

⎫⎬

⎭ d�

⎤

⎥⎦ d I (t)

=∑

I∈Hm

B I (xKm )d I (t) (22)

where Hm = {I |supp(xI ) ⊃ xKm }, AKm and �Km denote thearea and boundary of the sub-domain �Km .B I (xKm ) is thesmoothed gradient matrix defined as:

B I (xKm ) =⎡

⎢⎣∇x�w

I,x (xKm ) ∇x�θxI,x (xKm ) ∇x�

θyI,x (xKm )

∇y�wI,y(xKm ) ∇y�

θxI,y(xKm ) ∇y�

θyI,y(xKm )

2∇y�wI,x (xKm ) 2∇y�

θxI,x (xKm ) 2∇y�

θyI,x (xKm )

⎤

⎥⎦

(23)

with

∇ j�αI,i (xKm ) = 1

AKm

∫

�Km

�αI,i n j d�,

α = {w, θx , θy}, {i, j} = {x, y} (24)

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Comput Mech (2011) 48:47–63 51

Fig. 4 Comparison of dispersion properties for thin beam problemwith an = 1.3

Fig. 5 Comparison of dispersion properties for thin beam problemwith an = 2.5

Fig. 6 Comparison of dispersion properties for thin plate problem withk�x/π = 1

Fig. 7 Comparison of dispersion properties for thin plate problem withk�x/π = 0.5

Fig. 8 Comparison of dispersion properties for thin plate problem withk�x/π = 0.25

Fig. 9 A simply supported thin beam under harmonic point force atthe mid-span

3.3 Discrete equations

Introducing the sub-domain integration with the smoothedcurvature into the weak form of Eq. (20) yields:

NP∑

K=1

NSK∑

Km=1

[δwhT

(xcKm

)ρtwh(xcKm

) + δκhT(xKm )Dκh(xKm )

]AKm

+NP∑

K=1

NSK∑

Km=1

δW ext (xKm ) = 0 (25)

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52 Comput Mech (2011) 48:47–63

Fig. 10 Comparison of the center deflection for thin beam under har-monic point force: a deflection; b deflection error

where xcKm

is the centroid of the sub-domain �Km as shownin Fig. 2. The resulting matrix equation corresponding toEq. (25) is:

Md + Kd = f (26)

where M and K are the mass and stiffness matrices, f is theexternal force vector:

M = N PA

I,J=1[M I J ]; K = N P

AI,J=1

[K I J ]; f = N PA

I=1[f I ] (27)

with A denoting the standard global assembly operator. Themass and stiffness sub-matrices M I J , K I J and the force vec-tor f I are given as follows:

M I J =N P∑

K=1

N SK∑

Km=1

[�T

I

(xc

Km

)ρt� J

(xc

Km

)]AKm (28)

K I J =N P∑

K=1

N SK∑

Km=1

[B

TI (xKm )DB J (xKm )

]AKm (29)

Fig. 11 Comparison of the deflection at point J for thin beam underharmonic point force: a deflection; b deflection error

f I =N P∑

K=1

N SK∑

Km=1

[�T

I (xKm )c(xKi )]

AKm

+N Bint∑

L=1

[�T

I (xL )sn(xL ) − [∇� I (xL )]T n(xL )mnn(xL )]ωL

(30)

in which N Bint represent the number of integration samplingpoints xL ’s on the natural boundary and ωL is the correspond-ing weight. Commonly the trapezoidal rule can be used forthe boundary integration.

4 Dispersion analysis for semi-discrete formulation

4.1 Continuum analysis

Before analyzing the discrete behavior of the semi-discreteHRK meshfree equation, let’s first investigate the analytical

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Comput Mech (2011) 48:47–63 53

Fig. 12 Comparison of the deformation profile (t = 5) for thin beamunder harmonic point force: a deflection; b deflection error

Fig. 13 An irregular meshfree beam dicretization

frequency for the continuum thin plate problem. Recall theclassical deflection-based governing equation of thin plate:

ρtw + D∇4w = 0 (31)

where ∇4 is the bi-harmonic gradient operator. In dispersionanalysis the deflection can be assumed as:

w(x, y, t) = w0eι(kx cos θ+ky sin θ−ωt) (32)

where w0 is the wave amplitude, ι = √−1 is the complexnumber, k is the wave number, ω is the circular frequency,θ is the angle between the wave propagation direction and xaxis as shown in Fig. 3. Substituting Eq. (32) into Eq. (31)gives the continuum circular frequency ω:

ω = k2t

2

√E

3ρ(1 − ν2)(33)

Fig. 14 Comparison of the deformation profile (t = 5) for thin beamunder harmonic point force using irregular discretization: a deflection;b deflection error

Based on Eq. (33), the phase speed cp can be defined as:

cp = ω

k= kt

2

√E

3ρ(1 − ν2)(34)

In case of thin beam, the equation of motion becomes:

ρtw + E Id4w

dx4 = 0 (35)

with EI being the bending rigidity. The solution of Eq. (35)takes the form of:

w(x, t) = w0eι(kx−ωt) (36)

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54 Comput Mech (2011) 48:47–63

Fig. 15 Comparison of the center deflection for thin beam underharmonic point force using irregular discretization: a deflection;b deflection error

Similarly, plugging (36) into (35) yields the thin beam circu-lar frequency and phase speed:

ω = k2

√E I

ρ A(37)

cp = k

√E I

ρ A(38)

where A is the area of the beam cross section.

4.2 Discrete analysis

In meshfree discrete analysis, the semi-discrete equation ofmotion of Eq. (26) without the external force is considered:

Md + Kd = 0 (39)

Fig. 16 Comparison of the deflection at point J for thin beamunder harmonic point force using irregular discretization: a deflection;b deflection error

For convenience of expression, the particle number is rear-ranged as a two-index notation as shown in Fig. 3, i.e.,I = (i, j) and the nodal coefficient vector becomes:

d I (t) = d(i, j)(t) = d(xi , y j , t)

= {w(xi , y j , t), θx (xi , y j , t), θy(xi , y j , t)} (40)

Therefore the solution of Eq. (39) can be assumed as:

d(i, j)(t) =⎧⎨

⎩

w(xi , y j , t)θx (xi , y j , t)θy(xi , y j , t)

⎫⎬

⎭

=

⎧⎪⎨

⎪⎩

w0eι(kxi cos θ+ky j sin θ−ωh t)

θx0eι(kxi cos θ+ky j sin θ−ωh t)

θy0eι(kxi cos θ+ky j sin θ−ωh t)

⎫⎪⎬

⎪⎭(41)

where w0, θx0 and θy0 are the amplitudes of the waves asso-ciated with the deflection w, and the rotations θx and θy, ω

h

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Comput Mech (2011) 48:47–63 55

Fig. 17 Comparison of solutions for thin beam under constant steppoint force: a center deflection; b deformation profile (t = 5)

is the discrete circular frequency. Equation (41) also implies:

d(i, j)(t) = −(ωh)2d(i, j)(t) (42)

Moreover by using Eq. (41), for the neighbor meshfreeparticle xi+m = xi + m�x , y j+n = y j + n�y, one has:

d(i+m, j+n)(t) =⎧⎨

⎩

w(xi+m, y j+n, t)θx (xi+m, y j+n, t)θy(xi+m, y j+n, t)

⎫⎬

⎭

=⎧⎨

⎩

w(xi , y j , t)eι( km�x cos θ+kn�y sin θ)

θx (xi , y j , t)eι( km�x cos θ+kn�y sin θ)

θy(xi , y j , t)eι( km�x cos θ+kn�y sin θ)

⎫⎬

⎭

= eι( km�x cos θ+kn�y sin θ)d(i, j)(t) (43)

where it is noted that m and n could be positive or nega-tive integers. Thus with Eq. (43), one can readily obtain thefollowing relationship:

d(i+m, j+n)(t) = eι( km�x cos θ+kn�y sin θ) d(i, j)(t) (44)

Fig. 18 Refined comparison of the center deflection for thin beamunder constant step point force: a deflection; b deflection error

From Eq. (39), the semi-discrete equation of motionlocated at the node (xi , y j ) can be expressed as:∑

(i+m, j+n)∈G{M(i, j)(i+m, j+n) d(i+m, j+n)

+K(i, j)(i+m, j+n)d(i+m, j+n)} = 0 (45)

where G denote the group of nodes whose kernel functionscover the node xI = x(i, j):

G ={K =(i + m, j + n)|supp(xK ) ⊃ xI , I = (i, j)} (46)

Substituting Eq. (43) into Eq. (45) yields the following eigen-value problem:

[GK − (ωh)2GM ]d(i, j) = 0 (47)

where the 3 × 3 matrices GK and GM are given by:⎧⎪⎨

⎪⎩

GK = ∑(i+m, j+n)∈G

[K(i, j)(i+m, j+n)eι( km�x cos θ+kn�y sin θ)

]

GM = ∑(i+m, j+n)∈G

[M(i, j)(i+m, j+n)eι( km�x cos θ+kn�y sin θ)

] (48)

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56 Comput Mech (2011) 48:47–63

Fig. 19 Refined comparison of the deformation profile (t = 5) for thinbeam under constant step point force: a deflection; b deflection error

Fig. 20 A simply supported square plate under center harmonic pointforce

Clearly Eq. (47) implies the determinant of the coefficientmatrix should vanish:

det[GK − (ωh)2GM ] = 0 (49)

Fig. 21 Comparison of the center deflection error for square plateproblem

Fig. 22 Comparison of the deformed profile of the line y = 5 forsquare plate problem

Equation (49) can be further expanded as:

c6(ωh)6 + c4(ω

h)4 + c2(ωh)2 + c0 = 0 (50)

with

c6 = det(GM) (51)

c4 = −

∣∣∣∣∣∣∣

G K11 G M

12 G M13

G K21 G M

22 G M23

G K31 G M

32 G M33

∣∣∣∣∣∣∣−

∣∣∣∣∣∣∣

G M11 G K

12 G M13

G M21 G K

22 G M23

G M31 G K

32 G M33

∣∣∣∣∣∣∣

−

∣∣∣∣∣∣∣

G M11 G M

12 G K13

G M21 G M

22 G K23

G M31 G M

32 G K33

∣∣∣∣∣∣∣(52)

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Comput Mech (2011) 48:47–63 57

c2 =

∣∣∣∣∣∣∣

G M11 G K

12 G K13

G M21 G K

22 G K23

G M31 G K

32 G K33

∣∣∣∣∣∣∣+

∣∣∣∣∣∣∣

G K11 G M

12 G K13

G K21 G M

22 G K23

G K31 G M

32 G K33

∣∣∣∣∣∣∣

+

∣∣∣∣∣∣∣

G K11 G K

12 G M13

G K21 G K

22 G M23

G K31 G K

32 G M33

∣∣∣∣∣∣∣(53)

c0 = −det(

GK)

(54)

Equation (50) can be used to extract the desired real andphysical numerical circular frequency ωh .

For thin beam problem, following a similar procedure asthose of the thin plate described above, it can be shown thecharacteristic equation governing the numerical frequencyωh is:

c4(ωh)4 + c2(ω

h)2 + c0 = 0 (55)

where⎧⎪⎨

⎪⎩

GK = ∑(i+m)∈G

[Ki(i+m)eιkm�x

]

GM = ∑(i+m)∈G

[Mi(i+m)eιkm�x

] (56)

c4 = det(GM) (57)

c2 = −∣∣∣∣∣G K

11 G M12

G K21 G M

22

∣∣∣∣∣−∣∣∣∣∣G M

11 G K12

G M21 G K

22

∣∣∣∣∣ (58)

c0 = det(GK ) (59)

Thus the numerical frequencies can be obtained as:

ωh1 =

√√√√−c2 −√

c22 − 4c4c0

2c4(60)

ωh2 =

√√√√−c2 +√

c22 − 4c4c0

2c4(61)

The frequency ωh1 given by Eq. (60) is often referred as the

acoustic branch frequency and it is the desired physical fre-quency, while and the ωh

2 given by Eq. (61) is called theoptical branch frequency which comes from the numericaldiscretization [54].

After the numerical frequency ωh is obtained, a normal-ized phase speed cpn can be defined by the ratio betweenthe discrete frequency ωh and the continuum frequency dis-cussed earlier:

cpn = ωh

ω(62)

Obviously one would expect that cpn approaches unity as thedicscretization is refined. Thus cpn can serves as a useful toolto assess the frequency behavior of numerical schemes.

5 Transient analysis for full-discrete formulation

In the transient analysis, the time domain [0, T ] is dividedinto N time steps, i.e., t0 = 0, t1, . . . , tn, tn+1, . . . , tN = T,

within a typical one step algorithm one would like to updatethe field variables from tn to tn+1. The following Newmarkmethod is employed here to temporally discretize the semi-discrete equation of (26):

(i) Predictor phase:

{dn+1 = dn + (�tn)vn + (�tn)2

2 (1 − 2β)an

vn+1 = vn + (�tn)(1 − γ )an(63)

(ii) Corrector phase:

{dn+1 = dn+1 + β(�tn)2an+1

vn+1 = vn+1 + γ (�tn)an+1(64)

where �tn = tn+1 − tn, dn, vn and an denote the deflec-tion, velocity and acceleration vectors at time tn . β and γ areNewmark parameters. By choosing β = 0 and γ = 0.5 onget the central difference scheme that is used in this study.Substituting Eq. (64) into Eq. (26) that is evaluated at tn+1

gives the fully discretized equation of motion:

(M + β(�tn)2K)an+1 = f n+1 − Kdn+1 (65)

6 Results and discussions

6.1 Dispersive property of thin beam formulation

Consider a thin beam whose geometry and material prop-erties are: length L = 20, cross-section width b = 0.5,

height t = 0.5, density ρ = 2,800, Young’s modulus E =2 × 1011. It should be pointed out that the geometric andmaterial properties would not affect the analysis results sincethe normalized values will be presented. In the analysis ameshfree discretization with 21 particles is employed andthe HRK approximation is constructed based on the qua-dratic basis vector. The cubic B-spline function is used asthe kernel function. For comparison the Gauss integration(GI)-based HRK formulations are also listed here, where thenotation “GIm” denotes the mth order Gauss integration ineach dimension. The notation “SSCI2” represents the inte-gration using two sub-domains in each dimension. Figures 4and 5 show the dispersion results using different integrationmethods and support sizes, where the horizontal axis repre-sents the meshfree discretization and the vertical axis denotethe normalized phase speed. The cusp in Fig. 5 is due to the

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58 Comput Mech (2011) 48:47–63

Fig. 23 Comparison of the deflection error (t = 2.5 s) for square plate problem: a GI2; b GI6; c SSCI-Q4; d SSCI-T8

Fig. 24 Comparison of the deflection (t = 2.5 s) for square plate problem: a exact; b SSCI-T8

fact that the stiffness by GI2 is very close to being singu-lar under some circumstances. These results uniformly dem-onstrate that for thin beam problem the method of SSCI2exhibits superior dispersive characteristics compared withthe methods of GI2 and GI6. It is especially noted that fora smaller support size like an = 1.3, the present methodhas excellent dispersive behavior and this makes the methodmore favorable for practical analysis from the viewpoints ofboth efficiency and accuracy.

6.2 Dispersive property of thin plate formulation

Without loss of generality consider a square plate whosegeometry and material properties are: length a = 10,

thickness t = 0.05, density ρ = 8,000, Young’s mod-ulus E = 2 × 1011, and Poisson ratio υ = 0.3. Threemeshfree discretizations with 17 × 17 uniform distrib-uted particles are used in the analysis. Again the qua-dratic basis and the cubic B-spline kernel function with

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Comput Mech (2011) 48:47–63 59

Fig. 25 Irregular discretizations for square plate problem: a SSCI-Q4;b SSCI-T8

a normalized support size of 1.3 are employed in theHRK approximation. Figures 6, 7 and 8 show the com-parisons of the normalized phase speed for various inte-gration methods, where k�x/π = 1, 0.5, 0.25 implythat each wave length covers 3, 5 and 9 meshfree par-ticles, respectively. Four integration methods are inves-tigated, in which “SSCI-Q4” denotes that two by twoquadrilateral sub-domains are employed for the domainintegration and “SSCI-T8” represents that the integra-tion is carried out by using eight triangular sub-domains.The curvature smoothing sub-domains used for SSCI-T8can be readily obtained through further subdividing eachquadrilateral sub-domain in SSCI-Q4 into two triangularsub-domains. The results evince that SSCI-T8 show the bestdispersive property compared to other three methods such asSSCI-Q4, GI6 and GI2 regardless of the wave propagationdirections. On the other hand the performance of SSCI-Q4 isdirection-dependent and this approach only gives very goodnormalized phase speed as the wave propagation directioncoincides with the x or y axis, i.e., θ = 0◦ or θ = 90◦. Thus

Fig. 26 Comparison of the center deflection for square plate problemwith irregular discretization: a deflection; b deflection error

the SSCI-T8 formulation is the most preferable method forplate problems.

6.3 Transient analysis of simply supported beam underharmonic point force

As shown in Fig. 9, the simply supported beam is loaded atthe mid-span by a harmonic point force F(t) = F0sin(ωt)with ω = π and F0 = 10. The geometry and material prop-erties of the beam is: length L = 10, cross-section widthb = 1, height t = 1, density ρ = 2,500, Young’s modulusis E = 2 × 106. The analytic solution for this problem is[55]:

w(x, t) = 2F0

ρ AL

∞∑

i=1,3,5,...

[sin

(iπ

2

)sin(iπx/L)

ω2i − ω2

×[

sin(ωt) − ω

ωisin(ωi t)

]]; ωi = i2π2

L2

√E I

ρ A(66)

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60 Comput Mech (2011) 48:47–63

Fig. 27 Comparison of the deflection error (t = 5 s) for square plate problem with irregular discretization: a GI2; b GI6; c SSCI-Q4; d SSCI-T8

where A = bh is the area of the beam cross section.In the transient analysis firstly the beam is modeled by

11 evenly spaced meshfree particles. The HRK approxi-mation employs the quadratic basis vector and the cubicB-spline kernel function with a 1.3 normalized supportsize. The time step is set to be �t = 0.01 s. For con-venience the error = wh(x, t) − we(x, t) is defined tocompare the accuracy of various methods. The time his-tory of the center deflection and the deflectional errors ofthe points C and J as indicated in Fig. 9 are shown inFigs. 10 and 11 where it can be clearly seen that method ofSSCI2 gives the most favorable solution accuracy and GI2yields the largest solution errors. The deformed beam pro-files and the related solution errors at t = 5 are plotted inFig. 12. Once again the results support that SSCI2 performssuperiorly compared to GI6 and GI2. These observationsare consistent with the aforementioned dispersion analysisresults. To further demonstrate the effectiveness of the pres-ent method, an irregular discretization as shown in Fig. 13is employed in the analysis. The corresponding results arelisted in Figs. 14, 15 and 16, which clearly show that GI2is very sensitive to the spatial discretization and even fail-ure to predict the overall deformation trend, while SSCI2is very robust and still yields the most favorable solutionaccuracy.

6.4 Transient analysis of simply supported beam underconstant step point load

The simply supported beam described in the previous exam-ple is re-considered here but in this case the point force actingat the mid-span is a constant step excitation F0 = 10. Theanalytical solution for this problem is [55]:

w(x, t) = 2F0 L3

π4 E I

∞∑

i=1

[1

i4 sin(nπ

2

)sin

iπx

Lsin

iπ

2

(1 − cos ωi t)

]; ωi = i2π2

L2

√E I

ρ A(67)

In the transient analysis similar to the previous example,the beam is uniformly discretized with 11 meshfree parti-cles. Meanwhile the quadratic basis vector and the cubicB-spline kernel function with a 1.3 normalized support sizeare used construct the HRK shape function. The time inte-gration step is also �t = 0.01. The time history of the centerbeam deflection and the beam deformation profile are plottedin Fig. 17, where three methods are compared and it turnsout GI2 gives highly oscillating and instable results with pro-nounced error, while the results of GI6 and SSCI2 are quitecomparable. In Fig. 18, a refined comparison between the

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Comput Mech (2011) 48:47–63 61

GI6 and SSCI2 is given on the center deflection and relatederror, as shows SCCI2 yields more accurate solution. More-over, Fig. 19 gives the refined comparison of the deforma-tion profile at t = 5 between the GI6 and SSCI2, which alsostates that the method of SSCI2 produces much better solu-tion accuracy. The calculations performed at other points inthe domain confirm these findings as well.

6.5 Transient analysis of simply supported plate underharmonic point load

As shown in Fig. 20, a simply supported square plate is sub-jected to a sinusoidal concentrated force acting at the platecenter. The geometry and material parameters are: lengtha = 10, thickness t = 0.05, density ρ = 8,000, Young’smodulus E = 2 × 1011, Poisson ratio ν = 0.3. The forcecoefficient F0 = 100 and the force frequency � = π. Theanalytical solution for this problem is [55]:

w(x, y, t) =∞∑

m=1

∞∑

n=1

Wmn(x, y)ηmn(t) (68)

with

Wmn(x, y) = 2

a√

ρtsin(mπx

a

)sin(nπy

a

)(69)

ηmn(t) = 2F0(ω2

mn − �2)

a√

ρtsin(mπ

2

)sin(nπ

2

)

×(

sin�t − �

ωmnsinωmnt

)(70)

In the transient analysis a uniform discretization of 9 × 9meshfree particles is first employed. The quadratic basisfunction and the cubic B-spline function with normalizedsupport size of 1.3 are used to formulate the HRK approxi-mation. The time step used herein is �t = 0.01. Four meth-ods, such as GI2, GI6, SSCI-Q4 and SSCI-T8, are used tocompute the dynamic response of this plate problem. Thecenter deflection errors associated with various methods areshown in Fig. 21, which clearly states that the method ofSSCI-T8 gives the most accurate result and GI2 yields thelowest solution accuracy. Figure 22 lists the results of thedeformed profiles for the line y = 5 and they also supportthat SSCI-T8 performs superiorly compared to other meth-ods. The deflectional errors produced by various methods areshown in Fig. 23, where clearly SSCI-T8 gives the most accu-rate solution. The superior performance of SSCI-T8 is onceagain confirmed in Fig. 24 in which the solution of SSCI-T8compares favorably with the analytical solution. Meanwhilethe non-uniform meshfree plate discretizations as shown inFig. 25 are used to verify the proposed method. The resultsof center deflection and deformation profiles are shown inFigs. 26 and 27, which again reveals that the method SSCI-T8 performs superiorly compared with other methods.

7 Conclusions

A dispersion analysis of the HRK Galerkin meshfree formu-lation for thin beam and plate problems was presented. TheHRK meshfree thin beam and plate formulation was fulfilledby the SSCI in which the smoothed sub-domain curvaturewas adopted for the domain integration of Galerkin weakform. In the dispersion analysis both the nodal deflectionaland rotational variables were expressed by harmonic func-tions and then substituted into the semi-discrete equationof motion to obtain the characteristic equation. Thereafterthe characteristic equation was used to extract the numeri-cal circular frequency. Meanwhile the analytical continuumfrequencies for thin beam and plate problems were also dis-cussed in details. Thus the normalized phase speed couldbe obtained as the ratio between the discrete and continuumfrequencies. The dispersion analysis results revealed that themethod of SSCI-T8 gives the best dispersive properties com-pared to other methods such as GI2, GI6 and SSCI-Q4 for thinplate problem. It is noted that for thin beam problem SSCI-T8and SSCI-Q4 coincides and they are denoted SSCI2 methodtherein. The beam results supported that SSCI2 performssuperiorly compared to the GI2 and GI6 approaches. Sub-sequently the full-discrete equation of motion was obtainedby using the central difference time integration scheme.Several beam and plate examples further evinced that thetransient analysis results are consistent with those of the dis-persion analysis and the proposed method yields very favor-able solution accuracy.

Acknowledgments The financial support of this work by the NationalNatural Science Foundation of China (10972188), the FundamentalResearch Funds for the Central Universities of China (2010121073)and the Program for New Century Excellent Talents in Univer-sity from China Education Ministry (NCET-09–0678) is gratefullyacknowledged.

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